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I guess part of this depends on what you mean by the existence of an analytical solution...

If you mean "can it be written in terms of some convergent series of special functions?" then the answer is that nobody has found a set of special functions that are reasonably well-understood enough to be useful. That is, you could just proclaim a special function which solves the associated differential equation. This is what all special functions are, really. But if you don't know things like the eigenvalues, then this is rather pointless. The reason here is, ultimately, nobody's found a good representation for the solutions. Maybe you could -- if you could exactly solve Helium it would be quite the accomplishment!

If you mean "is there some complete set of commuting operators?" then this boils down to the integrability of the many-body problem. This has been a subject of interest for at least 200 years. This leads to a lot of interactions between the KAM theorem, quantum mechanics from a Lie algebraic perspective, regular versus chaotic dynamics, and a lot of really fascinating stuff. But the answer here might be "No, there is not a complete set of operators that have simultaneous eigenvectors with the Hamiltonian", or rather that there is no set $\{\mathcal{J}_i \}_N$ for $2N$ degrees of freedom such that

$[\mathcal{J}_i, \mathcal{H}] = 0 ~ \forall ~ i$

But there are approximate solutions. If you look in Sakurai for the helium atom, a variational approach works reasonably well at predicting the ground state energies including the inter-electron Coulomb repulsion. In THAT sense, there are analytical solutions. But all this depends on what you mean by "analytical solutions".